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$\begingroup$I don't see a question about basic definitions that could be answered by consulting any glossary or undergraduate text on graph theory (e.g. see en.wikipedia.org/wiki/Glossary_of_graph_theory) as being appropriate here, but maybe that's just me.$\endgroup$
– David EppsteinDec 10 '09 at 23:04

$\begingroup$I agree with Alex. Note that Strongly connected means "there is a route/path" instead of "there is an edge" between every two nodes.$\endgroup$
– user2079Nov 25 '09 at 20:24

$\begingroup$Alex, can you explain a bit more on the difference between a Connected Graph and a Complete Graph? It seems the only difference is that one uses path and the other uses edge. Aren't they the same? When you said for a Complete Graph, it's when: "are undirected graphs where there is an edge between every pair of nodes". Well, since it's an undirected graph then you can traverse both ways, hence why it's an "edge". But doesn't that mean the same as 'path'? I.e, there's a path between every two nodes that you can traverse between? So isn't that just the same as the definition of a Connected Graph$\endgroup$
– user18992Nov 3 '11 at 8:21

$\begingroup$what is the difference between a path and a route?$\endgroup$
– Cacho SantaOct 29 '13 at 0:56

$\begingroup$@cacho According to the answer, it seems route is commonly used for directed graph and path for undirected graphs.$\endgroup$
– Sumit GeraJan 21 '15 at 6:19

It is also important to remember the distinction between strongly connected and unilaterally connected. A directed graph is unilaterally connected if for any two vertices a and b, there is a directed path from a to b or from b to a but not necessarily both (although there could be). Strongly connected implies that both directed paths exist. This means that strongly connected graphs are a subset of unilaterally connected graphs.

And a directed graph is weakly connected if it's underlying graph is connected.